Stamps
Given a set of N stamp values (e.g., {1 cent, 3 cents}) and an upper limit K to the number of stamps that can fit on an envelope, calculate the largest unbroken list of postages from 1 cent to M cents that can be created.
For example, consider stamps whose values are limited to 1 cent and 3 cents; you can use at most 5 stamps. It's easy to see how to assemble postage of 1 through 5 cents (just use that many 1 cent stamps), and successive values aren't much harder:
- 6 = 3 + 3
- 7 = 3 + 3 + 1
- 8 = 3 + 3 + 1 + 1
- 9 = 3 + 3 + 3
- 10 = 3 + 3 + 3 + 1
- 11 = 3 + 3 + 3 + 1 + 1
- 12 = 3 + 3 + 3 + 3
- 13 = 3 + 3 + 3 + 3 + 1.
However, there is no way to make 14 cents of postage with 5 or fewer stamps of value 1 and 3 cents. Thus, for this set of two stamp values and a limit of K=5, the answer is M=13.
The most difficult test case for this problem has a time limit of 3 seconds.
PROGRAM NAME: stamps
INPUT FORMAT
| Line 1:
| Two integers K and N. K (1 <= K <= 200) is the total number of stamps that can be used. N (1 <= N <= 50) is the number of stamp values. |
| Lines 2..end:
| N integers, 15 per line, listing all of the N stamp values, each of which will be at most 10000. |
SAMPLE INPUT (file stamps.in)
5 2
1 3
OUTPUT FORMAT
| Line 1:
| One integer, the number of contiguous postage values starting at 1 cent that can be formed using no more than K stamps from the set. |
SAMPLE OUTPUT (file stamps.out)
13
Analysis
This problem is also a DP problem. Considering the minimum volumn of a bag that cannot be stored by the K stamps and N kinds of the given stamps, we can find the problem becomes to find the dynamic function to describe it. We call f[i] as the minimum number of stamps to get the value of i, which depends on the minimum of a set of f[i-value[j]]+1),((i-Value[j]>=0 j=1..Stamps).
In order to abbreviate functioning amount, find out the MaxValue, which is the maximum of set Value[i], and all of its multiples can be only fixed with i/MaxValue.
At last,the edge of the problem is when f[k]>maxuse and print k-1.
Code
/**//*
ID:braytay1
PROG:stamps
LANG:C++
*/
#include <iostream>
#include <fstream>
#define min(a,b) (a<b)?a:b
using namespace std;
int main()
{
ifstream fin("stamps.in");
ofstream fout("stamps.out");
int k1,n,max=-1;
int s[51];
short int f[2000000];
fin>>k1>>n;
for (int i=1;i<=n;i++) {fin>>s[i];max=(max>s[i])?max:s[i];}
if (max==1) {fout<<max*k1<<endl;return 0;}
memset(f,100,sizeof(f));
f[0]=0;
for (int j=1,k=max;k<=max*k1;k+=max,j++) f[k]=j;
int kk;
for (kk=1;kk<=max*k1;++kk){
if (kk%max==0) continue;
for (int j=1;j<=n;++j) {
if (kk-s[j]>=0)
if (f[kk-s[j]]+1<f[kk]) f[kk]=f[kk-s[j]]+1;
}
if (f[kk]>k1) break;
}
fout<<kk-1<<endl;
return 0;
}